Show that two lines are parallel if and only if for two distinct points $v_1$ and $v_2$ on the first line, and two distinct points $w_1$ and $w_2$ on the second line, the difference $w_1 – w_2$ is a multiple of $v_1 – v_2$.
I am wondering if my answer is rigorous enough and my thinking didn't skip any steps. I think it feels solid, but I also feel that there might be some circular reasoning going on too.
Let $u_1$ and $u_2$ be vectors defined as $u_1 = v_1 – v_2$ and $u_w = w_1 – w_2$. This makes $u_1$ and $u_2$ be vectors from $v_1$ to $v_2$ and $w_1$ to $w_2$ respectively. We can rewrite $u_1$ and $u_2$ so that these vectors are represented parametrically
$$\begin{align*}
u_1 &= tu_1 – v_2 \\
u_2 &= tu_2 – w_2
\end{align*}$$
To show that these two lines are parallel, then $u_1$ and $u_w$ must be linear combinations such that there are solutions for
$$\begin{align*}
u_1 &= tu_2 – v_2 \\
u_2 &= tu_1 – w_2
\end{align*}$$
Best Answer
I am sure that you will be allowed to assume that two non-zero vectors are parallel if and only if one is a multiple of the other.
You have correctly obtained a vector $v_1-v_2$ which is parallel to one line and $w_1-w_2$ which is parallel to the other line. You can therefore conclude that the lines are parallel if and only if these vectors are parallel i.e. if and only if one is a multiple of the other.